Applications of integration (Chapter 16) 441
Example 4 Self Tutor
Use
Zb
a
[yU¡yL]dx to find the area bounded by thex-axis and y=x^2 ¡ 2 x.
The curve cuts thex-axis when y=0
) x^2 ¡ 2 x=0
) x(x¡2) = 0
) x=0or 2
) thex-intercepts are 0 and 2.
Area=
Z 2
0
[yU¡yL]dx
=
Z 2
0
[0¡(x^2 ¡ 2 x)]dx
=
Z 2
0
(2x¡x^2 )dx
=
·
x^2 ¡
x^3
3
̧ 2
0
=
¡
4 ¡^83
¢
¡(0)
) the area is^43 units^2.
Example 5 Self Tutor
Find the area of the region enclosed by y=x+2and y=x^2 +x¡ 2.
y=x+2meets y=x^2 +x¡ 2
where x^2 +x¡2=x+2
) x^2 ¡4=0
) (x+ 2)(x¡2) = 0
) x=§ 2
Area=
Z 2
¡ 2
[yU¡yL]dx
=
Z 2
¡ 2
[(x+2)¡(x^2 +x¡2)]dx
=
Z 2
¡ 2
(4¡x^2 )dx
=
·
4 x¡
x^3
3
̧ 2
¡ 2
=
¡
8 ¡^83
¢
¡
¡
¡8+^83
¢
=10^23 units^2
) the area is 1023 units^2 :
2
y
O x
y=0U
y=x-2xL 2
-2^12
2
-2
y
O x
y=x+2
y=x +x-2 2
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